MinT - Best Known (103−59, 103, s)-Nets in Base 27

Best Known (103−59, 103, s)-Nets in Base 27

(103−59, 103, 166)-Net over F₂₇ — Constructive and digital

Digital (44, 103, 166)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 36, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (8, 67, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(103−59, 103, 288)-Net over F₂₇ — Digital

Digital (44, 103, 288)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(103−59, 103, 370)-Net in Base 27 — Constructive

(44, 103, 370)-net in base 27, using

t-expansion [i] based on (43, 103, 370)-net in base 27, using
- 5 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(103−59, 103, 48580)-Net in Base 27 — Upper bound on s

There is no (44, 103, 48581)-net in base 27, because

1 times m-reduction [i] would yield (44, 102, 48581)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 99 800583 418483 485933 842409 963889 995956 534626 129807 227528 111751 097579 805566 690070 636270 322671 110889 066689 921683 235610 950846 597002 640657 311424 749395 > 27¹⁰² [i]